The presented mathematical model of sets with incomplete information is based on L-valued sets in universes endowed with symmetric and transitive L-valued relations R where L is a complete and atomic Boolean algebra. Values R(x, x) express incomplete information about the presence of elements in universes. In addition, incomplete information about the equality of elements and membership relations of sets is modeled. The work introduces a logic for structures with incomplete information and preliminary results on ordered sets and concept lattices with incomplete information.The presented mathematical model of sets with incomplete information is based on L-valued sets in universes endowed with symmetric and transitive L-valued relations R where L is a complete and atomic Boolean algebra. Values R(x, x) express incomplete information about the presence of elements in universes. In addition, incomplete information about the equality of elements and membership relations of sets is modeled. The work introduces a logic for structures with incomplete information and preliminary results on ordered sets and concept lattices with incomplete information.